Methods for coding and decoding LDPC codes, and method for forming LDPC parity check matrix

ABSTRACT

Provided are methods for encoding and decoding low-density parity-check (LDPC) codes and a method for forming an LDPC parity check matrix. The method for forming the LDPC parity check matrix, includes the steps of: preparing a plurality of parity check matrixes; and selecting a parity check matrix having maximum performance from the prepared parity check matrixes, wherein the parity check matrix has a degree distribution G(x) that meets an equation, 
                 G   ⁡     (   x   )       =             ∑     k   =   2           d   i     -   1       ⁢     a   k     ⁢     x   k       +         ∑     k   =     d   i           d   max       ⁢   C   ⁢           ⁢     k     -   γ       ⁢     x   k     ⁢           ⁢   or   ⁢           ⁢     G   ⁡     (   x   )           =           ⁢         ∑     k   =   2         d   max       ⁢       C   ⁡     (     k   +   α     )         -   γ       ⁢     x   k           ,         
where, a k  is a parameter that corresponds to the probability that nodes of the graph have a degree k, C is a parameter that is determined by a normalization condition, G(1)=1, and α, γ are parameters that is optimized through numerical calculations. This may significantly decrease a time to discover an optimal code by using a power-law distribution with less parameters in designing the LDPC codes, as compared with numerical optimization in a global parameter space.

BACKGROUND

1. Field of the Invention

The present invention relates to methods for coding and decodinglow-density parity-check (LDPC) codes and a method for forming an LDPCparity check matrix, more particularly, to methods for coding anddecoding LDPC codes and a method for forming an LDPC parity check matrixusing a parity check matrix, which has a degree distribution conformingto a power-law function.

2. Discussion of Related Art

An LDPC code introduced first by Gallager in 1962 is a linear block codein which most of elements of a parity check matrix are zero. Relatedcontents are described in “R. G. Gallager, Low-Density Parity-Checkcode,” IRE Trans. Inform. Theory, vol. IT-8, pp. 21-28, January 1962.However, the code was forgotten for a long time because it was difficultto implement a decoder with technologies at that time, and then has beenrediscovered by Mackay, et al. Related contents are described in “D. J.C. MacKay, Good error correcting codes based on very sparse matrices,”IEEE Trans. Inform. Theory, vol. 45, pp. 399-431, March 1999. Iterativedecoding using sum-product (summation and multiplication) algorithmsallows such an LDPC code to hold excellent error recovery performance.In particular, it has been shown (ascertained) that if the LDPC code isconfigured to be not uniform, namely, an irregular LDPC code, it resultsin excellent performance over a turbo code. Related contents aredescribed in “T. J. Richardson, M. A. Shokrollashi, and R. L. Urbanke,Design of capacity approaching irregular low-density parity-checkcodes,” IEEE Trans. Inform. Theory, vol. 47, no. 2, pp. 619-637,February 2001. The performance of the LDPC code is generally determinedby a configuration of the parity check matrix.

The encoding of the LDPC code can be achieved by an equation of c=aG. Inthe equation, c means an encoded code vector, a is an informationvector, and G corresponds to a parity generation matrix. The paritygeneration matrix G and the parity check matrix H satisfy an equation,cH^(T)=aGH^(T)=O, for all codewords c or all information vectors a. Inthe equation, O means a zero vector. In FIG. 1, denoted are examples ofthe information vector a, the parity generation matrix G, the codevector c, the parity check matrix H, and the zero vector O.

The decoding of the LDPC codes may use a bipartite graph depicted inFIG. 2. The bipartite graph in FIG. 2 corresponds to the parity checkmatrix shown in FIG. 1. The bipartite graph is composed of check nodes110 and variable nodes 120, and the variable nodes 120 are composed ofinformation nodes 122 and parity nodes 124. The number of the checknodes 110 is equal to the number of rows in the parity check matrix, andthe number of the variable node 120 is equal to the number of columns inthe parity check matrix, namely, a length of the code vector. Lines thatconnect between the check node 110 and the variable node 120 are callededges 130. The edges insertions follow the positions of 1 s of theparity matrix. For example, since there are 1 s at the first, fourth,sixth and seventh columns of the first row in the parity check matrix Hof FIG. 1, the first check node of FIG. 2 is connected to the first,fourth, sixth and seventh variable nodes by the edges. Decoding of theLDPC codes is done by an iterative belief propagation via the edge nodesbetween the check nodes 110 and the variable nodes 120. Related contentsare described in “D. J. C MacKay, Good error correcting codes based onvery sparse matrices,” IEEE Trans. Inform. Theory, vol. 45. pp. 399-431,March 1999.

Design of a good connection structure of a bipartite graph, on which theLDPC code is based, is of prime importance in constructing the LDPCcodes. This is because the connection structure between nodes in thegraph directly determines the performance of the LDPC code. An LDPC codemay be represented by a bipartite graph consisting of variable nodes andcheck nodes, as described above. The number of edges connected to thenode is called the degree of the node. It is known that the performanceof codes formulated on the graph is determined by the degreedistribution of the graph on the assumption that the graph is a randomgraph having a tree structure and does not have a cycle therein. Giventhe degree distribution of the graph, the performance of the graph canbe calculated by the density evolution method developed by Richardson,et al. Related contents are described in “T. J. Richardson, R. L.Urbanke, IEEE Trans. Inform. Theory 47, 599, 2001” and “T. J.Richardson, M. A. Shokrollahi, R. L. Urbanke, IEEE Trans. Inform. Theory47, 619, 2001.”

A conventional method for designing the connection structure of thegraph, namely, constructing a parity check matrix used for the LDPCcodes includes the degree optimization though numerical calculation. Onthe other hand, codes that provide mathematic expressions for the degreedistribution include Tornado code developed by Luby, et al., aright-regular sequence of Shokrollahi, and the like. However, suchformation of the LDPC codes according to the prior art requires massivenumerical computation and finding an optimal solution is not a feasibletask, as is similar to a typical global optimization problem.

SUMMARY OF THE INVENTION

The present invention is directed to a method for forming a parity checkmatrix that is used for LDPC codes, an encoder and a decoder for theLDPC code using the same, in which a time required for the codeoptimization is significantly reduced.

According to the first aspect of the present invention, there isprovided a method for encoding LDPC codes, comprised of the steps of:preparing a parity generation matrix; and forming a code vector bymultiplying an information vector, which is source data to be coded, bythe parity generation matrix, wherein the parity generation matrix meetsan equation, O=aGH^(T), where, O is a zero vector, a is an arbitraryinformation vector, G is a parity generation matrix, and H is a paritycheck matrix, and the parity check matrix has a degree distribution G(x)given by the following equation:

${{G(x)} = {{{\overset{d_{i} - 1}{\sum\limits_{k = 2}}a_{k}x^{k}} + {\overset{d_{\max}}{\sum\limits_{k = d_{i}}}C\; k^{- \gamma}x^{k}\mspace{14mu}{or}\mspace{11mu}{G(x)}}} = \;{\overset{d_{\max}}{\sum\limits_{k = 2}}{C\left( {k + \alpha} \right)}^{- \gamma}x^{k}}}},$where, a_(k) is a parameter that corresponds to the probability thatnodes of the graph have a degree k, C is a parameter that is determinedby a normalization condition, G(1)=1, and α, γ are parameters that isoptimized through numerical calculations.

According to the second aspect of the present invention, there isprovided a method for decoding an LDPC code, comprising the steps of:preparing a parity check matrix; and obtaining an information vectorfrom the parity check matrix and a received code vector, wherein theparity check matrix has a degree distribution G(x) given by thefollowing equation,

${{G(x)} = {{{\overset{d_{i} - 1}{\sum\limits_{k = 2}}a_{k}x^{k}} + {\overset{d_{\max}}{\sum\limits_{k = d_{i}}}C\; k^{- \gamma}x^{k}\mspace{14mu}{or}\mspace{11mu}{G(x)}}} = \;{\overset{d_{\max}}{\sum\limits_{k = 2}}{C\left( {k + \alpha} \right)}^{- \gamma}x^{k}}}},$where, a_(k) is a parameter that corresponds to the probability thatnodes of the graph have a degree k, C is a parameter that is determinedby a normalization condition, G(1)=1, and α, γ are parameters that isoptimized through numerical calculations.

According to the third aspect of the present invention, there isprovided a method for forming an LDPC parity check matrix, comprisingthe steps of: preparing a plurality of parity check matrixes; andselecting a parity check matrix having maximum performance from theprepared parity check matrixes, wherein: the parity check matrix has adegree distribution G(x) given by the following equation:

${{G(x)} = {{{\overset{d_{i} - 1}{\sum\limits_{k = 2}}a_{k}x^{k}} + {\overset{d_{\max}}{\sum\limits_{k = d_{i}}}C\; k^{- \gamma}x^{k}\mspace{14mu}{or}\mspace{14mu}{G(x)}}} = {\overset{d_{\max}}{\sum\limits_{k = 2}}{C\left( {k + \alpha} \right)}^{- \gamma}x^{k}}}},$where, a_(k) is a parameter that corresponds to the probability thatnodes of the graph have a degree k, C is a parameter that is determinedby a normalization condition, G(1)=1, and α, γ are parameters that isoptimized through numerical calculations.

Hereinafter, theoretic background of the present invention will bedescribed.

The present invention utilizes a scale-free network, which has a degreedistribution conforming to a power-law function, as a basis graph forLDPC codes. Related explanations are described in “R. Albert, A. L.Barabasi, Rev. Mod. Phys. 74, 47, 2002.” In many cases, actual complexnetworks have an inter-node connection distribution conforming to apower-law function. Such a network is called a scale-free network, andan example of the scale-free network includes Internet and World WideWeb, as well as a society network such as a co-writer network, andbiological networks such as a metabolism network, a protein network, andthe like.

The degree distribution G(x) of a typical graph may be simply expressedby Equation 1:

$\begin{matrix}{{G(x)} = {\overset{d_{\max}}{\sum\limits_{k = 2}}a_{k}x^{k}}} & \left\lbrack {{Equation}\mspace{14mu} 1} \right\rbrack\end{matrix}$

In the above equation, a_(k) corresponds to the probability that nodesof the graph have a degree k, and thus G(1)=1. The summation of theequation begins with k=2, since nodes having a degree smaller than 2 donot contribute to error correction.

In the case of the scale-free network, the degree distribution G(x) isin the form of Equation 2.

[Equation 2]

$\begin{matrix}{{G(x)} = {{\overset{d_{i} - 1}{\sum\limits_{k = 2}}a_{k}x^{k}} + {\overset{d_{\max}}{\sum\limits_{k = d_{i}}}C\; k^{- \gamma}x^{k}}}} & \left\lbrack {{Equation}\mspace{14mu} 2} \right\rbrack\end{matrix}$where, a_(k) is a parameter that corresponds to the probability thatnodes of the graph have a degree k, C is a parameter that is determinedby a normalization condition, G(1)=1, and α, γ are parameters that isoptimized through numerical calculations, and nodes having a degree ofk≧d_(l) have a degree distribution conforming to the power-law function.Note that in the above equation, the degree distribution of nodes havinga degree smaller than d_(l) may not conform to the power-law function.In a normal actual scale-free network, its degree distribution conformsto the power-law function at large degrees, but not at small degrees.Likewise, even in the case of the scale-free network used in the presentinvention, there is a likelihood that the degree distribution may notexactly conform to power-law distribution in the region where the degreeis small.

Although the degree distribution in Equation 2 may be used as it is, adegree distribution in a more simplified form represented by Equation 3may be used:

$\begin{matrix}{{{G(x)} = {\overset{d_{\max}}{\sum\limits_{k = 2}}C\left( {k + \alpha} \right)^{- \gamma}x^{k}}}\;} & \left\lbrack {{Equation}\mspace{14mu} 3} \right\rbrack\end{matrix}$

There is an advantage that the degree distribution represented byEquation 3 has fewer parameters than those of the degree distributionrepresented by Equation 2. That is, in the case of Equation 2,optimization is difficult since it includes a large number of parameters(a2, a3, . . . , a_(dl−1)) while in the case of Equation 3, actual codeoptimization may be performed in a two-dimensional parameter spacecomposed of parameters α and γ since the equation includes only threeparameters (α, γ, C), in which C can be determined by a normalizationcondition, namely, G(1)=1.

The check nodes of the graph have a degree distribution defined byEquation 4:F(x)=bx ^(k)+(1−b)x ^(k+1)  [Equation 4]

In Equation 4, F(x) meets the normalization condition F(1)=1 as G(x),and b and k are parameters that are automatically determined when adegree distribution of variable nodes is given.

Encoding of the code may be performed using a belief propagationalgorithm, which is commonly used in decoding the LDPC codes. Throughnumerical calculations, optimized α and γ or b and k, giving the maximumperformance can be found.

The performance of the LDPC codes on a binary erasure channel may becalculated by the density evolution method developed by Luby,Richardson, et al., as described below. Hereinafter, the densityevolution method will be discussed. Assuming that δ is erasureprobability of a given erasure channel, consider LDPC codes having adegree distribution given as λ(x)=Σλ_(k)x^(k−1) and ρ(x)=Σρ_(k)x^(k−1).In this equation, λ_(k) is the probability that a connecting lineselected arbitrarily in the graph becomes a connecting line connected toa variable node having a degree k, and meets a relationship equation ofλ_(k)=Cka_(k) with Equation 5. ρ_(k) is probability that a connectingline selected arbitrarily in the graph becomes a connecting lineconnected to a check node having the degree k, and meets a relationshipequation of ρ_(k)=Ckb_(k) with Equation 5. Here, C is a normalizationconstant, yielding Σλ_(k)=1. If decoding is performed using the beliefpropagation algorithm, the messages delivered between the variable nodesand the check nodes meet the following equation:x _(l) =x ₀λ(1−ρ(1−x _(l−1)))  [Equation 5]

In the above equation, x₁ is erasure message probability at the l-thiteration, x₀ is an initial value of this probability and x₀=δ. If x_(l)converges into 0 when l increases, it means that errors occurred in thedata transmitted through a given channel are successfully corrected. Themaximum value δ* of the erasure probability leading to successfuldecoding is an index representing the performance of the code. δ* alwayshas a value smaller than 1−R with respect to a given code rate R.

The performance of the scale-free network based code calculated by theabove-stated density evolution method is denoted in FIG. 3. In FIG. 3, aselected code rate, R=0.5. It can be seen that δ* increases as themaximum degree d_(max) increases, and δ* substantially reaches 1−R,which is a theoretical limit value of the performance, at a largerd_(max).

To recognize the performance more specifically, comparison was madebetween the performance of the code according to the present inventionand the performance of the Tornado code developed by Luby, et al., andthe result of the comparison is given in Table 1.

TABLE 1 Codes Tornado on SFN codes d_(max) γ α δ* <k> δ* <k> 9 1.347−1.473 0.47875 2.97 0.44546 3 16 1.788 −1.102 0.48633 3.30 0.46950 3.528 2.024 −0.868 0.49163 3.57 0.48235 4 47 2.088 −0.755 0.49477 3.880.48960 4.5 79 2.084 −0.753 0.49689 4.24 0.49380 5 133 2.080 −0.7410.49810 4.60 0.49628 5.5 222 2.086 −0.712 0.49862 4.94 0.49776 6 3682.081 −0.698 0.49895 5.31 0.49865 6.5 610 2.076 −0.691 0.49920 5.680.49918 7 1009 2.073 −0.687 0.49931 6.02 0.49951 7.5

As can be seen from Table 1, codes on scale-free networks (SFN) basedcode exhibits excellent performance over the Tornado codes in an areawhere d_(max) is smaller than or equal to 610. In Table 1, <k> is anaverage degree of the variable node, which represents physicalcomplexity of the graph. From the comparison at the same d_(max) orperformance, it can be seen that the physical complexity of thescale-free network based codes is lower than that of the Tornado codes,as shown in Table 1.

The scale-free network based LDPC code has another advantage that aniteration number required for decoding convergence is small. FIGS. 4 aand 4 b show an iteration number required for the decoding convergenceof the scale-free network based LDPC codes and the Tornado codes. InFIG. 4 a shows an iteration number required for the decoding convergencewhen dmax=610, and FIG. 4 b shows an iteration number required for thedecoding convergence when dmax=1009. Further, a dotted line representsan iteration number required for the decoding convergence of the Tornadocodes, and a solid line represents an iteration number required for thedecoding convergence of the scale-free network based LDPC codes. Whendmax=610, the iteration number of the scale-free network based codes issmaller than that of the Tornado codes in all erasure probabilityregions, and even when dmax=1009, the iteration number of the codessuggested by the present invention is smaller than that of the Tornadocodes over a broad area except for a section where δ is close to δ*.

Finally, the scale-free network based codes have a feature that it cantake all integer values for d_(max). at an arbitrary code rate, which isa property that the Tornado codes do not posses; the Tornado codes cantake only a particular d_(max) at a given code rate. This propertyallows for more flexibility in formulating the scale-free network basedcodes, compared with the Tornado codes.

BRIEF DESCRIPTION OF THE DRAWINGS

The above and other features and advantages of the present inventionwill become more apparent to those of ordinary skill in the art bydescribing in detail preferred embodiments thereof with reference to theattached drawings in which:

FIG. 1 is a diagram for explaining LDPC code encoding;

FIG. 2 is a diagram for explaining LDPC code decoding;

FIG. 3 is a graph illustrating the performance of scale-free networkbased codes calculated by the density evolution method;

FIG. 4 a and FIG. 4 b are graphs showing an iteration number requiredfor decoding convergence of scale-free network based LDPC codes andTornado codes;

FIG. 5 is a flow diagram illustrating a method for coding scale-freenetwork based LDPC codes according to the first embodiment of thepresent invention;

FIG. 6 is a flow diagram illustrating a method for decoding scale-freenetwork based LDPC codes according to the second embodiment of thepresent invention; and

FIG. 7 is a flow diagram illustrating a method for forming a scale-freenetwork based LDPC parity check matrix according to the third embodimentof the present invention.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

Hereinafter, preferred embodiments of the present invention will bedescribed in detail with reference to the accompanying drawings.However, the embodiments of the present invention may be changed intoseveral other forms, and it should not be construed that the scope ofthe present invention is limited to the embodiments described in detailbelow. The embodiments of the present invention are intended to explainthe present invention more completely to those skilled in the art.

FIG. 5 is a flow diagram illustrating a method for coding scale-freenetwork based LDPC codes according to the first embodiment of thepresent invention.

Referring to FIG. 5, a method for encoding LDPC codes includes a stepS11 of preparing a parity generation matrix; and a step S12 of forming acode vector by multiplying an information vector, which is data desiredto be coded, by the parity generation matrix. Here, the paritygeneration matrix satisfies the following Equation 6:O=aGH^(T),  [Equation 6]

where, O means a zero vector, a means an information vector, G means aparity generation matrix, and H means a parity check matrix. The paritycheck matrix has a degree distribution G(x) given by the followingEquation 7 or 8:

$\begin{matrix}{{G(x)} = {{\overset{d_{i} - 1}{\sum\limits_{k = 2}}{a_{k}x^{k}}} + {\overset{d_{\max}}{\sum\limits_{k = d_{i}}}{C\; k^{- \gamma}x^{k}}}}} & \left\lbrack {{Equation}\mspace{14mu} 7} \right\rbrack\end{matrix}$where, G(x) is the degree distribution of the parity check matrix, a_(k)is a parameter that corresponds to the probability that nodes of thegraph have a degree k, C is a parameter that is determined by anormalization condition, G(1)=1, and α, γ are parameters that isoptimized through numerical calculations.

$\begin{matrix}{{{G(x)} = {\overset{d_{\max}}{\sum\limits_{k = 2}}C\left( {k + \alpha} \right)^{- \gamma}x^{k}}}\;} & \left\lbrack {{Equation}\mspace{14mu} 8} \right\rbrack\end{matrix}$where, G(x) is the degree distribution of the parity check matrix, a_(k)is a parameter that corresponds to the probability that nodes of thegraph have a degree k, C is a parameter that is determined by anormalization condition, G(1)=1, and α, γ are parameters that isoptimized through numerical calculations.The parity check matrix may be a parity check matrix having maximumperformance selected from a plurality of parity check matrixes that meetthe degree distribution of Equation 7 or 8. The performance of theparity check matrix may be calculated using the density evolutionmethod.

FIG. 6 is a flow diagram illustrating a method for decoding scale-freenetwork based LDPC codes according to the second embodiment of thepresent invention.

Referring to FIG. 6, a method for decoding LDPC codes includes a stepS21 of preparing a parity check matrix, and a step S22 of obtaining aninformation vector from the parity check matrix and a code vector. Here,the degree distribution of the parity check matrix can meet Equation 7or 8.

The parity check matrix may be a parity check matrix having maximumperformance selected from a plurality of parity check matrixes that meetthe degree distribution of Equation 7 or 8. The performance of theparity check matrix may be calculated using the density evolutionmethod. Further, the step of obtaining the information vector from theparity check matrix and the code vector may be implemented using acommonly used belief propagation algorithm.

FIG. 7 is a flow diagram illustrating a method for forming a scale-freenetwork based LDPC parity check matrix according to the third embodimentof the present invention.

Referring to FIG. 7, a method for forming an LDPC parity check matrixincludes a step S31 of preparing a plurality of parity check matrixes;and a step S32 of selecting a parity check matrix having maximumperformance from the prepared parity check matrixes. Here, the paritycheck matrix may have a degree distribution that meets Equation 7 or 8.

The performance of the parity check matrix may be calculated using thedensity evolution method.

First, the present invention can significantly decrease a computationaltime required to find an optimal code in designing the LDPC codes sinceit uses a power-law distribution with a smaller number of parameters, ascompared with an exhaustive numerical search for global optimization.Second, the present invention has an advantage that it exhibits moreexcellent performance for an erasure channel when the code has a maximumdegree smaller than about 1000, as compared with the Tornado code. Thepresent invention may be more useful in a situation where the physicalcomplexity and maximum degree of the codes are restricted.

Third, the present invention has an advantage that it is capable ofdesigning the code at all integer values of maximum degree for anarbitrarily given code rate. Thus, the present invention allows for aflexible code design without a restriction on the maximum size of thedegree.

Fourth, the present invention has an advantage that it may be applied tovarious channels other than the erasure channel as they are.

Although the present invention has been described in detail by way ofthe detailed embodiments, the present invention is not limited to theembodiments, and it will be apparent that variations and modificationsmay be made to the present invention by those skilled in the art withoutdeparting from the technical spirit of the present invention.

1. A method for encoding LDPC codes, comprising the steps of: preparinga parity generation matrix; and forming a code vector by multiplying aninformation vector, which is data desired to be coded, by the paritygeneration matrix, wherein the parity generation matrix meets thefollowing Equation:O=aGH^(T), where, O is a zero vector, a is an arbitrary informationvector, G is the parity generation matrix, and H is a parity checkmatrix, and wherein the parity check matrix has a degree distributionG(x) that meets the following Equation:${{{\overset{d_{i} - 1}{\sum\limits_{k = 2}}a_{k}x^{k}} + {\overset{d_{\max}}{\sum\limits_{k = d_{i}}}{C\; k^{- \gamma}x^{k}}}}\;,}\mspace{11mu}$where, a_(k) is a parameter that corresponds to the probability thatnodes of the graph have a degree k, C is a parameter that is determinedby a normalization condition, G(1)=1, and γ is a parameter that isoptimized through numerical calculations.
 2. The method according toclaim 1, further comprising step of: calculating a performance of theparity check matrix by a density evolution method.
 3. The methodaccording to claim 2, wherein the parity check matrix is selected from aplurality of parity check matrixes that meet the degree distribution. 4.A method for coding LDPC codes, comprising the steps of: preparing aparity generation matrix; and forming a code vector by multiplying aninformation vector, which is data desired to be coded, by the paritygeneration matrix, wherein the parity generation matrix meets thefollowing Equation: O=aGH^(T), where, O is a zero matrix, a is anarbitrary information vector, G is the parity generation matrix, and His a parity check matrix, and wherein the parity check matrix has adegree distribution G(x) that meets the following Equation:${G(x)} = {\underset{k = 2}{\sum\limits^{d_{\max}}}{{C\left( {k + \alpha} \right)}^{- \gamma}x^{k}}}$where, C is a parameter that is determined by a normalization condition,G(1)=1, and aα, γ are parameters that are optimized through numericalcalculations.
 5. The method according to claim 4, further comprisingstep of: calculating a performance of the parity check matrix by adensity evolution method.
 6. The method according to claim 5, whereinthe parity check matrix is selected from a plurality of parity checkmatrixes that meet the degree distribution.
 7. A method for decodingLDPC codes, comprising the steps of: preparing a parity check matrix;and obtaining an information vector from the parity check matrix and areceived code vector, wherein the parity check matrix has a degreedistribution G(x) that meets the following Equation:${G(x)} = {{\underset{k = 2}{\sum\limits^{d_{i} - 1}}{a_{k}x^{k}}} + {\underset{k = d_{i}}{\sum\limits^{d_{\max}}}{Ck}^{- \gamma}x^{k}}}$where, a_(k) is a parameter that corresponds to the probability thatnodes of the graph have a degree k, C is a parameter that is determinedby a normalization condition, G(1)=1 and γ is a parameter that isoptimized through numerical calculations.
 8. The method according toclaim 7, further comprising step of: calculating a performance of theparity check matrix by a density evolution method.
 9. The methodaccording to claim 8, wherein the parity check matrix is selected from aplurality of parity check matrixes that meet the degree distribution.10. A method for decoding LDPC codes, comprising the steps of: preparinga parity check matrix; and obtaining an information vector from theparity check matrix and a received code vector, wherein the parity checkmatrix has a degree distribution G(x) that meets the following Equation:${G(x)} = {\underset{k = 2}{\sum\limits^{d_{\max}}}{{C\left( {k + \alpha} \right)}^{- \gamma}x^{k}}}$where, C is a parameter that is determined by a normalization condition,G(1)=1, and γ is a parameter that is optimized through numericalcalculations.
 11. The method according to claim 10, further comprisingstep of: calculating a performance of the parity check matrix by adensity evolution method.
 12. The method according to claim 11, whereinthe parity check matrix is selected from a plurality of parity checkmatrixes that meet the degree distribution.
 13. A method for forming anLDPC parity check matrix, comprising the steps of: preparing a pluralityof parity check matrixes; and selecting a parity check matrix havingmaximum performance of the prepared parity check matrixes, wherein theparity check matrix has a degree distribution G(x) that meets thefollowing Equation:${G(x)} = {{\underset{k = 2}{\sum\limits^{d_{i} - 1}}{a_{k}x^{k}}} + {\underset{k = d_{i}}{\sum\limits^{d_{\max}}}{Ck}^{- \gamma}x^{k}}}$where, a_(k) is a parameter that corresponds to the probability thatnodes of the graph have a degree k, C is a parameter that is determinedby a normalization condition, G(1)=1, and α, γ are parameters that isoptimized through numerical calculations.
 14. A method for forming anLDPC parity check matrix, comprising the steps of: preparing a pluralityof parity check matrixes; and selecting a parity check matrix havingmaximum performance of the prepared parity check matrixes, wherein theparity check matrix has a degree distribution G(x) that meets thefollowing Equation:${G(x)} = {\underset{k = 2}{\sum\limits^{d_{\max}}}{{C\left( {k + \alpha} \right)}^{- \gamma}x^{k}}}$where, C is a parameter that is determined by a normalization condition,G(1)=1, and α, γ are parameters that is optimized through numericalcalculations.